Computational analysis of looping of a large family of highly bent DNA by LacI

Linearized Bayesian inference for Young's modulus parameter field in an elastic model of slender structures

The deformations of several slender structures at nano-scale are conceivably sensitive to their non-homogenous elasticity. Owing to their small scale, it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments. Molecular dynamics simulations can provide an alternative or additional source of data. However, the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations. In this paper, we formulate an inverse problem governed by a linear elastic model in a Bayesian inference framework. To make the problem tractable, we use a Gaussian approximation of the posterior probability distribution that results from the Bayesian solution of the inverse problem of inferring Young's modulus parameter fields from available data. The performance of the computational framework is demonstrated using two representative loading scenarios, one involving cantilever bending and the other involving stretching of a helical rod (an intrinsically curved structure). The results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data. We also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions.

DNA buckling in bacteriophage cavities as a mechanism to aid virus assembly

While relatively simple biologically, bacteriophages are sophisticated biochemical machines that execute a precise sequence of events during virus assembly, DNA packaging, and ejection. These stages of the viral life cycle require intricate coordination of viral components whose structures are being revealed by single molecule experiments and high resolution (cryo-electron microscopy) reconstructions. For example, during packaging, bacteriophages employ some of the strongest known molecular motors to package DNA against increasing pressure within the viral capsid shell. Located upstream of the motor is an elaborate portal system through which DNA is threaded. A high resolution reconstruction of the portal system for bacteriophage ϕ29 reveals that DNA buckles inside a small cavity under large compressive forces. In this study, we demonstrate that DNA can also buckle in other bacteriophages including T7 and P22. Using a computational rod model for DNA, we demonstrate that a DNA buckle can initiate and grow within the small confines of a cavity under biologically-attainable force levels. The forces of DNA-cavity contact and DNA-DNA electrostatic repulsion ultimately limit cavity filling. Despite conforming to very different cavity geometries, the buckled DNA within T7 and P22 exhibits near equal volumetric energy density (∼1kT/nm(3)) and energetic cost of packaging (∼22kT). We hypothesize that a DNA buckle creates large forces on the cavity interior to signal the conformational changes to end packaging. In addition, a DNA buckle may help retain the genome prior to tail assembly through significantly increased contact area with the portal.

Tethered particle motion reveals that LacI·DNA loops coexist with a competitor-resistant but apparently unlooped conformation

The lac repressor protein (LacI) efficiently represses transcription of the lac operon in Escherichia coli by binding to two distant operator sites on the bacterial DNA and causing the intervening DNA to form a loop. We employed single-molecule tethered particle motion to observe LacI-mediated loop formation and breakdown in DNA constructs that incorporate optimized operator binding sites and intrinsic curvature favorable to loop formation. Previous bulk competition assays indirectly measured the loop lifetimes in these optimized DNA constructs as being on the order of days; however, we measured these same lifetimes to be on the order of minutes for both looped and unlooped states. In a range of single-molecule DNA competition experiments, we found that the resistance of the LacI-DNA complex to competitive binding is a function of both the operator strength and the interoperator sequence. To explain these findings, we present what we believe to be a new kinetic model of loop formation and DNA competition. In this proposed new model, we hypothesize a new unlooped state in which the unbound DNA-binding domain of the LacI protein interacts nonspecifically with nonoperator DNA adjacent to the operator site at which the second LacI DNA-binding domain is bound.

Role of microscopic flexibility in tightly curved DNA

The genetic material in living cells is organized into complex structures in which DNA is subjected to substantial contortions. Here we investigate the difference in structure, dynamics, and flexibility between two topological states of a short (107 base pair) DNA sequence in a linear form and a covalently closed, tightly curved circular DNA form. By employing a combination of all-atom molecular dynamics (MD) simulations and elastic rod modeling of DNA, which allows capturing microscopic details while monitoring the global dynamics, we demonstrate that in the highly curved regime the microscopic flexibility of the DNA drastically increases due to the local mobility of the duplex. By analyzing vibrational entropy and Lipari–Szabo NMR order parameters from the simulation data, we propose a novel model for the thermodynamic stability of high-curvature DNA states based on vibrational untightening of the duplex. This novel view of DNA bending provides a fundamental explanation that bridges the gap between classical models of DNA and experimental studies on DNA cyclization, which so far have been in substantial disagreement.

Structural ensemble and dynamics of toroidal-like DNA shapes in bacteriophage ϕ29 exit cavity

In the bacteriophage ϕ29, DNA is packed into a preassembled capsid from which it ejects under high pressure. A recent cryo-EM reconstruction of ϕ29 revealed a compact toroidal DNA structure (30-40 basepairs) lodged within the exit cavity formed by the connector-lower collar protein complex. Using multiscale models, we compute a detailed structural ensemble of intriguing DNA toroids of various lengths, all highly compatible with experimental observations. In particular, coarse-grained (elastic rod) and atomistic (molecular dynamics) models predict the formation of DNA toroids under significant compression, a largely unexplored state of DNA. Model predictions confirm that a biologically attainable compressive force of 25 pN sustains the toroid and yields DNA electron density maps highly consistent with the experimental reconstruction. The subsequent simulation of dynamic toroid ejection reveals large reactions on the connector that may signal genome release.

The pattern and evolution of looped gene bendability

Gene looping, defined as the physical interaction between the promoter and terminator regions of a RNA polymerase II-transcribed gene, is widespread in yeast and mammalian cells. Gene looping has been shown to play important roles in transcription. Gene-loop formation is dependent on regulatory proteins localized at the 5' and 3' ends of genes, such as TFIIB. However, whether other factors contribute to gene looping remains to be elucidated. Here, we investigated the contribution of intrinsic DNA and chromatin structures to gene looping. We found that Saccharomyces cerevisiae looped genes show high DNA bendability around middle and 3/4 regions in open reading frames (ORFs). This bendability pattern is conserved between yeast species, whereas the position of bendability peak varies substantially among species. Looped genes in human cells also show high DNA bendability. Nucleosome positioning around looped ORF middle regions is unstable. We also present evidence indicating that this unstable nucleosome positioning is involved in gene looping. These results suggest a mechanism by which DNA bendability and unstable nucleosome positioning could assist in the formation of gene loops.

Rationally designed coiled-coil DNA looping peptides control DNA topology

Artificial DNA looping peptides were engineered to study the roles of protein and DNA flexibility in controlling the geometry and stability of protein-mediated DNA loops. These LZD (leucine zipper dual-binding) peptides were derived by fusing a second, C-terminal, DNA-binding region onto the GCN4 bZip peptide. Two variants with different coiled-coil lengths were designed to control the relative orientations of DNA bound at each end. Electrophoretic mobility shift assays verified formation of a sandwich complex containing two DNAs and one peptide. Ring closure experiments demonstrated that looping requires a DNA-binding site separation of 310 bp, much longer than the length needed for natural loops. Systematic variation of binding site separation over a series of 10 constructs that cyclize to form 862-bp minicircles yielded positive and negative topoisomers because of two possible writhed geometries. Periodic variation in topoisomer abundance could be modeled using canonical DNA persistence length and torsional modulus values. The results confirm that the LZD peptides are stiffer than natural DNA looping proteins, and they suggest that formation of short DNA loops requires protein flexibility, not unusual DNA bendability. Small, stable, tunable looping peptides may be useful as synthetic transcriptional regulators or components of protein–DNA nanostructures.

Interplay of protein and DNA structure revealed in simulations of the lac operon

The E. coli Lac repressor is the classic textbook example of a protein that attaches to widely spaced sites along a genome and forces the intervening DNA into a loop. The short loops implicated in the regulation of the lac operon suggest the involvement of factors other than DNA and repressor in gene control. The molecular simulations presented here examine two likely structural contributions to the in-vivo looping of bacterial DNA: the distortions of the double helix introduced upon association of the highly abundant, nonspecific nucleoid protein HU and the large-scale deformations of the repressor detected in low-resolution experiments. The computations take account of the three-dimensional arrangements of nucleotides and amino acids found in crystal structures of DNA with the two proteins, the natural rest state and deformational properties of protein-free DNA, and the constraints on looping imposed by the conformation of the repressor and the orientation of bound DNA. The predicted looping propensities capture the complex, chain-length-dependent variation in repression efficacy extracted from gene expression studies and in vitro experiments and reveal unexpected chain-length-dependent variations in the uptake of HU, the deformation of repressor, and the folding of DNA. Both the opening of repressor and the presence of HU, at levels approximating those found in vivo, enhance the probability of loop formation. HU affects the global organization of the repressor and the opening of repressor influences the levels of HU binding to DNA. The length of the loop determines whether the DNA adopts antiparallel or parallel orientations on the repressor, whether the repressor is opened or closed, and how many HU molecules bind to the loop. The collective behavior of proteins and DNA is greater than the sum of the parts and hints of ways in which multiple proteins may coordinate the packaging and processing of genetic information.

LacI-DNA-IPTG loops: equilibria among conformations by single-molecule FRET

The E. coli Lac repressor (LacI) tetramer binds simultaneously to a promoter-proximal DNA binding site (operator) and an auxiliary operator, resulting in a DNA loop, which increases repression efficiency. Induction of the lac operon by allolactose reduces the affinity of LacI for DNA, but induction does not completely prevent looping in vivo. Our previous work on the conformations of LacI loops used a hyperstable model DNA construct, 9C14, that contains a sequence directed bend flanked by operators. Single-molecule fluorescence resonance energy transfer (SM-FRET) on a dual fluorophore-labeled LacI-9C14 loop showed that it adopts a single, stable, high-FRET V-shaped LacI conformation. Ligand-induced changes in loop geometry can affect loop stability, and the current work assesses loop population distributions for LacI-9C14 complexes containing the synthetic inducer IPTG. SM-FRET confirms that the high-FRET LacI-9C14 loop is only partially destabilized by saturating IPTG. LacI titration experiments and FRET fluctuation analysis suggest that the addition of IPTG induces loop conformational dynamics and re-equilibration between loop population distributions that include a mixture of looped states that do not exhibit high-efficiency FRET. The results show that repression by looping even at saturating IPTG should be considered in models for regulation of the operon. We propose that persistent DNA loops near the operator function biologically to accelerate rerepression upon exhaustion of inducer.

FRET studies of a landscape of Lac repressor-mediated DNA loops

DNA looping mediated by the Lac repressor is an archetypal test case for modeling protein and DNA flexibility. Understanding looping is fundamental to quantitative descriptions of gene expression. Systematic analysis of LacI•DNA looping was carried out using a landscape of DNA constructs with lac operators bracketing an A-tract bend, produced by varying helical phasings between operators and the bend. Fluorophores positioned on either side of both operators allowed direct Förster resonance energy transfer (FRET) detection of parallel (P1) and antiparallel (A1, A2) DNA looping topologies anchored by V-shaped LacI. Combining fluorophore position variant landscapes allows calculation of the P1, A1 and A2 populations from FRET efficiencies and also reveals extended low-FRET loops proposed to form via LacI opening. The addition of isopropyl-β-d-thio-galactoside (IPTG) destabilizes but does not eliminate the loops, and IPTG does not redistribute loops among high-FRET topologies. In some cases, subsequent addition of excess LacI does not reduce FRET further, suggesting that IPTG stabilizes extended or other low-FRET loops. The data align well with rod mechanics models for the energetics of DNA looping topologies. At the peaks of the predicted energy landscape for V-shaped loops, the proposed extended loops are more stable and are observed instead, showing that future models must consider protein flexibility.

DNA modeling reveals an extended lac repressor conformation in classic in vitro binding assays

Protein-mediated DNA looping, such as that induced by the lactose repressor (LacI) of Escherichia coli, is a well-known gene regulation mechanism. Although researchers have given considerable attention to DNA looping by LacI, many unanswered questions about this mechanism, including the role of protein flexibility, remain. Recent single-molecule observations suggest that the two DNA-binding domains of LacI are capable of splaying open about the tetramerization domain into an extended conformation. We hypothesized that if recent experiments were able to reveal the extended conformation, it is possible that such structures occurred in previous studies as well. In this study, we tested our hypothesis by reevaluating two classic in vitro binding assays using a computational rod model of DNA. The experiments and computations evaluate the looping of both linear DNA and supercoiled DNA minicircles over a broad range of DNA interoperator lengths. The computed energetic minima align well with the experimentally observed interoperator length for optimal loop stability. Of equal importance, the model reveals that the most stable loops for linear DNA occur when LacI adopts the extended conformation. In contrast, for DNA minicircles, optimal stability may arise from either the closed or the extended protein conformation depending on the degree of supercoiling and the interoperator length.

A multiscale dynamic model of DNA supercoil relaxation by topoisomerase IB

In this study, we report what we believe to be the first multiscale simulation of the dynamic relaxation of DNA supercoils by human topoisomerase IB (topo IB). We leverage our previous molecular dynamics calculations of the free energy landscape describing the interaction between a short DNA fragment and topo IB. Herein, this landscape is used to prescribe boundary conditions for a computational, elastodynamic continuum rod model of a long length of supercoiled DNA. The rod model, which accounts for the nonlinear bending, twisting, and electrostatic interaction of the (negatively charged) DNA backbone, is extended to include the hydrodynamic drag induced by the surrounding physiological buffer. Simulations for a 200-bp-long DNA supercoil in complex with topo IB reveal a relaxation timescale of ∼0.1-1.0 μs. The relaxation follows a sequence of cascading reductions in the supercoil linking number (Lk), twist (Tw), and writhe (Wr) that follow companion cascading reductions in the supercoil elastic and electrostatic energies. The novel (to our knowledge) multiscale modeling method may enable simulations of the entire experimental setup that measures DNA supercoiling and relaxation via single molecule magnetic trapping.

Conformation and mechanical properties of closed diblock fibers

We analyze the conformations and mechanical properties of closed diblock fibers. In our model, the length fraction of each component and the total fiber length are controlled by tunable chemical potentials. Our formalism can describe fibers in which one block is a bare polymer while the other is an adsorbed protein-filament complex; these blocks maintain different bending rigidities and spontaneous curvatures. We analytically calculate the shape of two-component polymers for all values of the material parameters and chemical potentials. Our results yield a complete analytical description of all possible two-component polymer conformations, a phase portrait detailing the parameter spaces in which these shapes occur, and the identification of spontaneous transitions between shapes driven by environmental changes.

Quantitative methods for measuring DNA flexibility in vitro and in vivo

The double-helical DNA biopolymer is particularly resistant to bending and twisting deformations. This property has important implications for DNA folding in vitro and for the packaging and function of DNA in living cells. Among the outstanding questions in the field of DNA biophysics are the underlying origin of DNA stiffness and the mechanisms by which DNA stiffness is overcome within cells. Exploring these questions requires experimental methods to quantitatively measure DNA bending and twisting stiffness both in vitro and in vivo. Here, we discuss two classical approaches: T4 DNA ligase-mediated DNA cyclization kinetics and lac repressor-mediated DNA looping in Escherichia coli. We review the theoretical basis for these techniques and how each can be applied to quantitate biophysical parameters that describe the DNA polymer. We then show how we have modified these methods and applied them to quantitate how apparent DNA physical properties are altered in vitro and in vivo by sequence-nonspecific architectural DNA-binding proteins such as the E. coli HU protein and eukaryotic HMGB proteins.

Gene repression by minimal lac loops in vivo

The inflexibility of double-stranded DNA with respect to bending and twisting is well established in vitro. Understanding apparent DNA physical properties in vivo is a greater challenge. Here, we exploit repression looping with components of the Escherichia coli lac operon to monitor DNA flexibility in living cells. We create a minimal system for testing the shortest possible DNA repression loops that contain an E. coli promoter, and compare the results to prior experiments. Our data reveal that loop-independent repression occurs for certain tight operator/promoter spacings. When only loop-dependent repression is considered, fits to a thermodynamic model show that DNA twisting limits looping in vivo, although the apparent DNA twist flexibility is 2- to 4-fold higher than in vitro. In contrast, length-dependent resistance to DNA bending is not observed in these experiments, even for the shortest loops constraining <0.4 persistence lengths of DNA. As observed previously for other looping configurations, loss of the nucleoid protein heat unstable (HU) markedly disables DNA looping in vivo. Length-independent DNA bending energy may reflect the activities of architectural proteins and the structure of the DNA topological domain. We suggest that the shortest loops are formed in apical loops rather than along the DNA plectonemic superhelix.

Looping probabilities of elastic chains: a path integral approach

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.